Chapter 4 Data Structures and Types of Variables

4.1 Data require structure and context

Descriptive statistics are concerned with the presentation, organization and summary of data, as suggested in Norman and Streiner (2014). This includes various methods of organizing and graphing data to get an idea of what those data can tell us.

As Vittinghoff et al. (2012) suggest, the nature of the measurement determines how best to describe it statistically, and the main distinction is between numerical and categorical variables. Even this is a little tricky - plenty of data can have values that look like numerical values, but are just numerals serving as labels.

As Bock, Velleman, and De Veaux (2004) point out, the truly critical notion, of course, is that data values, no matter what kind, are useless without their contexts. The Five W’s (Who, What [and in what units], When, Where, Why, and often How) are just as useful for establishing the context of data as they are in journalism. If you can’t answer Who and What, in particular, you don’t have any useful information.

In general, each row of a data frame corresponds to an individual (respondent, experimental unit, record, or observation) about whom some characteristics are gathered in columns (and these characteristics may be called variables, factors or data elements.) Every column / variable should have a name that indicates what it is measuring, and every row / observation should have a name that indicates who is being measured.

4.2 A New NHANES Adult Sample

In previous work, we spent some time with a sample from the National Health and Nutrition Examination. Now, by changing the value of the set.seed function which determines the starting place for the random sampling, and changing some other specifications, we’ll generate a new sample describing 500 adult subjects who completed the 2011-12 version of the survey when they were between the ages of 21 and 64.

Note also that what is listed in the NHANES data frame as Gender should be more correctly referred to as sex. Sex is a biological feature of an individual, while Gender is a social construct. This is an important distinction, so I’ll change the name of the variable. I’m also changing the names of three other variables, to create Race, SBP and DBP.

# library(NHANES) # NHANES package/library of functions, data

nh_temp <- NHANES %>%
    filter(SurveyYr == "2011_12") %>%
    filter(Age >= 21 & Age < 65) %>%
    mutate(Sex = Gender, Race = Race3, SBP = BPSysAve, DBP = BPDiaAve) %>%
    select(ID, Sex, Age, Race, Education, BMI, SBP, DBP, Pulse, PhysActive, Smoke100, SleepTrouble, HealthGen)

set.seed(431002) 
# use set.seed to ensure that we all get the same random sample 

nh_adults <- sample_n(nh_temp, size = 500) 

nh_adults

# A tibble: 500 x 13
      ID Sex     Age Race  Education   BMI   SBP   DBP Pulse PhysActive
   <int> <fct> <int> <fct> <fct>     <dbl> <int> <int> <int> <fct>     
 1 64427 male     37 White College ~  36.5   111    72    56 Yes       
 2 63788 fema~    40 White High Sch~  18.2   115    74   102 Yes       
 3 66874 fema~    31 White Some Col~  27.2    95    52    98 Yes       
 4 69734 male     26 White College ~  20.6   137    75    74 Yes       
 5 70409 male     44 White High Sch~  29.2   112    71    62 Yes       
 6 68961 fema~    64 White College ~  24.2   123    70    80 Yes       
 7 62616 fema~    37 Asian 8th Grade  19.3   109    73    82 No        
 8 70130 male     42 Black High Sch~  31.2   119    71    62 No        
 9 71218 male     33 White College ~  27.7   110    67    68 Yes       
10 69181 fema~    37 White 8th Grade  25     114    74    82 Yes       
# ... with 490 more rows, and 3 more variables: Smoke100 <fct>,
#   SleepTrouble <fct>, HealthGen <fct>

The data consist of 500 rows (observations) on 13 variables (columns). Essentially, we have 13 pieces of information on each of 500 adult NHANES subjects who were included in the 2011-12 panel.

4.2.1 Summarizing the Data’s Structure

We can identify the number of rows and columns in a data frame or tibble with the dim function.

dim(nh_adults)

[1] 500  13

The str function provides a lot of information about the structure of a data frame or tibble.

str(nh_adults)

Classes 'tbl_df', 'tbl' and 'data.frame':   500 obs. of  13 variables:
 $ ID          : int  64427 63788 66874 69734 70409 68961 62616 70130 71218 69181 ...
 $ Sex         : Factor w/ 2 levels "female","male": 2 1 1 2 2 1 1 2 2 1 ...
 $ Age         : int  37 40 31 26 44 64 37 42 33 37 ...
 $ Race        : Factor w/ 6 levels "Asian","Black",..: 5 5 5 5 5 5 1 2 5 5 ...
 $ Education   : Factor w/ 5 levels "8th Grade","9 - 11th Grade",..: 5 3 4 5 3 5 1 3 5 1 ...
 $ BMI         : num  36.5 18.2 27.2 20.6 29.2 24.2 19.3 31.2 27.7 25 ...
 $ SBP         : int  111 115 95 137 112 123 109 119 110 114 ...
 $ DBP         : int  72 74 52 75 71 70 73 71 67 74 ...
 $ Pulse       : int  56 102 98 74 62 80 82 62 68 82 ...
 $ PhysActive  : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 2 1 1 2 2 ...
 $ Smoke100    : Factor w/ 2 levels "No","Yes": 1 2 1 1 2 2 1 1 1 2 ...
 $ SleepTrouble: Factor w/ 2 levels "No","Yes": 1 2 1 1 1 1 1 1 1 2 ...
 $ HealthGen   : Factor w/ 5 levels "Excellent","Vgood",..: 2 3 3 1 3 2 3 3 3 2 ...

To see the first few observations, use head, and to see the last few, try tail…

tail(nh_adults, 5) # shows the last five observations in the data set

# A tibble: 5 x 13
     ID Sex     Age Race  Education   BMI   SBP   DBP Pulse PhysActive
  <int> <fct> <int> <fct> <fct>     <dbl> <int> <int> <int> <fct>     
1 69692 male     50 Black 9 - 11th~  22.7   132    82    60 No        
2 66472 male     61 White Some Col~  41.3   141    77    62 No        
3 71456 male     21 Mexi~ 9 - 11th~  26.7   113    66    78 No        
4 71420 fema~    54 Mexi~ 9 - 11th~  32.5   126    69    68 No        
5 63617 male     29 White College ~  23.2   105    72    76 Yes       
# ... with 3 more variables: Smoke100 <fct>, SleepTrouble <fct>,
#   HealthGen <fct>

4.2.2 What are the variables?

The variables we have collected are described in the brief table below³.

Variable	Description	Sample Values
ID	a numerical code identifying the subject	64427, 63788
Sex	sex of subject (2 levels)	male, female
Age	age (years) at screening of subject	37, 40
Race	reported race of subject (6 levels)	White, Asian
Education	educational level of subject (5 levels)	College Grad, High School
BMI	body-mass index, in kg/m²	36.5, 18.2
SBP	systolic blood pressure in mm Hg	111, 115
DBP	diastolic blood pressure in mm Hg	72, 74
Pulse	60 second pulse rate in beats per minute	56, 102
PhysActive	Moderate or vigorous-intensity sports?	Yes, No
Smoke100	Smoked at least 100 cigarettes lifetime?	Yes, No
SleepTrouble	Told a doctor they have trouble sleeping?	Yes, No
HealthGen	Self-report general health rating (5 lev.)	Vgood, Good

The levels for the multi-categorical variables are:

Race: Mexican, Hispanic, White, Black, Asian, or Other.
Education: 8th Grade, 9 - 11th Grade, High School, Some College, or College Grad.
HealthGen: Excellent, Vgood, Good, Fair or Poor.

4.3 Types of Variables

4.3.1 Quantitative Variables

Variables recorded in numbers that we use as numbers are called quantitative. Familiar examples include incomes, heights, weights, ages, distances, times, and counts. All quantitative variables have measurement units, which tell you how the quantitative variable was measured. Without units (like miles per hour, angstroms, yen or degrees Celsius) the values of a quantitative variable have no meaning.

It does little good to be promised a salary of 80,000 a year if you don’t know whether it will be paid in Euros, dollars, yen or Estonian kroon.
You might be surprised to see someone whose age is 72 listed in a database on childhood diseases until you find out that age is measured in months.
Often just seeking the units can reveal a variable whose definition is challenging - just how do we measure “friendliness”, or “success,” for example.
Quantitative variables may also be classified by whether they are continuous or can only take on a discrete set of values. Continuous data may take on any value, within a defined range. Suppose we are measuring height. While height is really continuous, our measuring stick usually only lets us measure with a certain degree of precision. If our measurements are only trustworthy to the nearest centimeter with the ruler we have, we might describe them as discrete measures. But we could always get a more precise ruler. The measurement divisions we make in moving from a continuous concept to a discrete measurement are usually fairly arbitrary. Another way to think of this, if you enjoy music, is that, as suggested in Norman and Streiner (2014), a piano is a discrete instrument, but a violin is a continuous one, enabling finer distinctions between notes than the piano is capable of making. Sometimes the distinction between continuous and discrete is important, but usually, it’s not.
- The nh_adults data includes several quantitative variables, specifically Age, BMI, SBP, DBP and Pulse.
- We know these are quantitative because they have units: Age in years, BMI in kg/m², the BP measurements in mm Hg, and Pulse in beats per minute.
- Depending on the context, we would likely treat most of these as discrete given that are measurements are fairly crude (this is certainly true for Age, measured in years) although BMI is probably continuous in most settings, even though it is a function of two other measures (Height and Weight) which are rounded off to integer numbers of centimeters and kilograms, respectively.
It is also possible to separate out quantitative variables into ratio variables or interval variables. An interval variable has equal distances between values, but the zero point is arbitrary. A ratio variable has equal intervals between values, and a meaningful zero point. For example, weight is an example of a ratio variable, while IQ is an example of an interval variable. We all know what zero weight is. An intelligence score like IQ is a different matter. We say that the average IQ is 100, but that’s only by convention. We could just as easily have decided to add 400 to every IQ value and make the average 500 instead. Because IQ’s intervals are equal, the difference between and IQ of 70 and an IQ of 80 is the same as the difference between 120 and 130. However, an IQ of 100 is not twice as high as an IQ of 50. The point is that if the zero point is artificial and moveable, then the differences between numbers are meaningful but the ratios between them are not. On the other hand, most lab test values are ratio variables, as are physical characteristics like height and weight. A person who weighs 100 kg is twice as heavy as one who weighs 50 kg; even when we convert kg to pounds, this is still true. For the most part, we can treat and analyze interval or ratio variables the same way.
- Each of the quantitative variables in our nh_adults data can be thought of as ratio variables.
Quantitative variables lend themselves to many of the summaries we will discuss, like means, quantiles, and our various measures of spread, like the standard deviation or inter-quartile range. They also have at least a chance to follow the Normal distribution.

4.3.2 Qualitative (Categorical) Variables

Qualitative or categorical variables consist of names of categories. These names may be numerical, but the numbers (or names) are simply codes to identify the groups or categories into which the individuals are divided. Categorical variables with two categories, like yes or no, up or down, or, more generally, 1 and 0, are called binary variables. Those with more than two-categories are sometimes called multi-categorical variables.

When the categories included in a variable are merely names, and come in no particular order, we sometimes call them nominal variables. The most important summary of such a variable is usually a table of frequencies, and the mode becomes an important single summary, while the mean and median are essentially useless.
- In the nh_adults data, Race is clearly a nominal variable with multiple unordered categories.
The alternative categorical variable (where order matters) is called ordinal, and includes variables that are sometimes thought of as falling right in between quantitative and qualitative variables.
- Examples of ordinal multi-categorical variables in the nh_adults data include the Education and HealthGen variables.
- Answers to questions like “How is your overall physical health?” with available responses Excellent, Very Good, Good, Fair or Poor, which are often coded as 1-5, certainly provide a perceived order, but a group of people with average health status 4 (Very Good) is not necessarily twice as healthy as a group with average health status of 2 (Fair).
Sometimes we treat the values from ordinal variables as sufficiently scaled to permit us to use quantitative approaches like means, quantiles, and standard deviations to summarize and model the results, and at other times, we’ll treat ordinal variables as if they were nominal, with tables and percentages our primary tools.
Note that all binary variables may be treated as ordinal, or nominal.
- Binary variables in the nh_adults data include Sex, PhysActive, Smoke100, SleepTrouble. Each can be thought of as either ordinal or nominal.

Lots of variables may be treated as either quantitative or qualitative, depending on how we use them. For instance, we usually think of age as a quantitative variable, but if we simply use age to make the distinction between “child” and “adult” then we are using it to describe categorical information. Just because your variable’s values are numbers, don’t assume that the information provided is quantitative.

References

Norman, Geoffrey R., and David L. Streiner. 2014. Biostatistics: The Bare Essentials. Fourth. People’s Medical Publishing House.

Vittinghoff, Eric, David V. Glidden, Stephen C. Shiboski, and Charles E. McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. Second. Springer-Verlag, Inc. http://www.biostat.ucsf.edu/vgsm/.

Bock, David E., Paul F. Velleman, and Richard D. De Veaux. 2004. Stats: Modelling the World. Boston MA: Pearson Addison-Wesley.

Descriptions are adapted from the ?NHANES help file. Remember that what NHANES lists as Gender is captured here as Sex, and similarly Race3, BPSysAve and BPDiaAve from NHANES are here listed as Race, SBP and DBP.↩